+603 When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.
When \frac{1}{288} is expressed in base 25, is it terminating or repeating?
+921 First, let's convert each number in the fraction to base 25.
1 always stays the same. We want to convert 288 into base 25. We get that
\((BD)_{25} = (11 × 25^1) + (13 × 25^0) = (288)_{10}\)
So now, we have that \(\frac{1}{288}_{10} = \frac{1}{BD}_{25}\)
Doing some base division, we find that \(\frac{1}{BD}_{25} = 0.\overline{0.02468ACEGIKN}\)
As shown, the decimals keep repeating, so 1/288 in base 25 is a repeating decimal.
So our answer is repeating.
Thanks! :)
